greatest at new and full moon, and least at the first and third quarter.*

Tycho fixed the maximum of this inequality at 40' 30". The value which results from modern observations is 39' 30".

122. We have already two epicycles, or one epicycle and an eccentric, to explain the first two inequalities: by the introduction of another epicycle or eccentric, the variation also might have been brought into the system; but Tycho adopted a different method:† like Ptolemy, he employed an eccentric for the evection, but for the first or elliptic inequality he employed a couple of epicycles, and this complicated combination, which it is needless further to describe, represented the change of distance better than Ptolemy's.

To introduce the variation, he imagined the centre of the larger epicycle to librate backwards and forwards on the eccentric, to an extent of 40′ on each side of its mean position; this mean place itself advancing uniformly along the eccentric with the moon's mean motion in anomaly; and the libration was so adjusted, that the moon was in her mean place at syzygy and quadrature, and at her furthest distance from it in the octants, the period of a complete libration being half a synodical revolution.

*It appears that Mohammed-Aboul-Wefa-al-Bouzdjani, an Arabian astronomer of the tenth century, who resided at Cairo, and observed at Bagdad in '975, discovered a third inequality of the moon, in addition to the two ex'pounded by Ptolemy, the equation of the centre and the evection. This 'third inequality, the variation, is usually supposed to have been discovered 'by Tycho Brahé, six centuries later...... In an almagest of Aboul-Wefa, a 'part of which exists in the Royal Library at Paris, after describing the two 'inequalities of the moon, he has a Section IX., "Of the third anomaly of the 'moon called Muhazal or Prosneusis"......But this discovery of Aboul-Wefa appears to have excited no notice among his contemporaries and followers; at 'least it had been long quite forgotten, when Tycho Brahé rediscovered the same lunar inequality.' Whewell's Hist. of Inductive Sciences, vol. i. p. 243. For a full description of Tycho's hypothesis, see Delambre, Hist. de l'Ast. Mod., tom. I. p. 162, and An Account of the Astronomical Discoveries of Kepler, by Robert Small, p. 135.

DISCOVERY OF ANNUAL EQUATION.

Annual Equation.

99

123. Tycho Brahé was also the discoverer of the fourth inequality, called the annual equation. This was connected with the anomalistic motion of the sun, and did not, like the previous inequalities, depend on the position of the moon in her orbit.

Having calculated the position of the moon corresponding to any given time, he found that the observed place was behind her computed one while the sun moved from perigee to apogee, and before it in the other half year.

Tycho did not state this distinctly, but he made a correction which, though wrong in quantity and applied in an indirect manner, shewed that he had seen the necessity and understood the law of this inequality.

He did not try to represent it by any new eccentric or epicycle, but he increased by (8m. 13s.) sin(sun's anomaly) the time which had served to calculate the moon's place; thus assuming that the true place, after that interval, would agree with the calculated one. Now, as the moon moves through 4′ 30′′ in 8m. 13s., it is clear that adding (8m. 13s.) sin(sun's anomaly) to the time is the same thing as subtracting (4′ 30′′) sin(sun's anomaly) from the calculated longitude, which was therefore the correction virtually introduced by Tycho.† Modern observations shew the coefficient to be 11' 9".

We have seen, Art. (75), how this inequality may be inferred from our equations.

Reduction.

124. The next inequality in longitude which we have to consider, is not an inequality in the same sense as the foregoing;

* That is, the equation of time which he used for the moon differed by that quantity from that used for the sun.

+ Horrocks (1639) made the correction in the same manner as Tycho, but so increased it that the corresponding coefficient was 11′ 51′′ instead of 4' 30". Flamsteed was the first to apply the correction to the longitude instead of the time.

that is, it does not arise from any irregularity in the motion of the moon herself in her orbit, but simply because that orbit is not in the same plane as that in which the longitudes are reckoned, so that even a regular motion in the one would be necessarily irregular when referred to the other. Thus if NMn (fig. 10) be the moon's orbit and TNm the ecliptic, and if M the moon be referred to the ecliptic by the great circle Mm perpendicular to it, then MN and mN are 0°, 90°, 180°, 270°, and 360° simultaneously, but they differ for all intermediate values: the difference between them is called the reduction.

The difference between the longitude of the node and that of the moon in her orbit being known, that is the side NM of the right-angled spherical triangle NMm, and also the angle N the inclination of the two orbits, the side Nm may be calculated by the rules of spherical trigonometry, and the difference between it and NM, applied with a proper sign to the longitude in the orbit, gives the longitude in the ecliptic.

Tycho was the first to make a table of the reduction instead of calculating the spherical triangle. His formula was

reduction tan sin2L – tan* sin 4L,

=

where I is the inclination of the orbit and L the longitude of the moon diminished by that of the node.

k2

The first term corresponds with the term - 1 sin2(gpt—y) of the expression for 0.

Latitude of the Moon.

125. That the moon's orbit is inclined to the ecliptic was known to the earliest astronomers, from the the non-recurrence of eclipses at every new and full moon; and it was also known, since the eclipses did not always take place in the same part of the heavens, that the line of nodes represented by Nn (fig. 10) has a retrograde motion on the ecliptic, N moving towards T.

Hipparchus fixed the inclination of the moon's orbit to the ecliptic at 5°, which value he obtained by observing the greatest

CHANGE OF INCLINATION.

101

distance at which she passes to the north or south of some star known to be in or very near the ecliptic, as for instance the bright star Regulus; and by comparing the recorded eclipses from the times of the Chaldean astronomers down to his own, he found that the line of nodes goes round the ecliptic in a retrograde direction in about 18 years.

This result is indicated in our expression for the value of the latitude by the term k sin(g0 − y), as we have shewn Art. (78).

126. Tycho Brahé further discovered that the inclination of the lunar orbit to the ecliptic was not a constant quantity of 5° as Hipparchus had supposed, but that it had a mean value of 5° 8', and ranged through 9′ 30′′ on each side of this, the least inclination 4° 58' occurring when the node was in syzygy, and the greatest 5° 17′ being attained when the node was in quadrature.*

He also found that the retrograde motion of the node was not uniform: the mean and true position of the node agreed very well when they were in syzygy or quadrature, but they were 1° 46′ apart in the octants.

By referring to Art. (80), we shall see that these corrections, introduced by Tycho Brahé, correspond to the second term of our expression for s.

Since Hipparchus could observe the moon with accuracy only in the eclipses, at which time the node is in or near syzygy,

* Ebn Jounis, an Arabian astronomer (died A.D. 1008), whose works were translated about 30 years since by Mons. Sedillot, states that the inclination of the moon's orbit had been often observed by Aboul-Hassan-Alyben-Amajour about the year 918, and that the results he had obtained were generally greater than the 5° of Hipparchus, but that they varied considerably.

Ebn Jounis adds, however, that he himself had observed the inclination several times and found it 5° 3', which leads us to infer that he always observed in similar circumstances, for otherwise a variation of nearly 23' could scarcely have escaped him. See Delambre, Hist. de l'Ast. du Moyen Age, p. 139.

The mean value of the inclination is 5° 8′ 55·46′′,—the extreme values are 4° 57′ 22′′ and 5° 20′ 6′′.

The mean daily motion of the line of nodes is 3′ 10.64", or one revolution in 6793.39 days, or 18y. 218d. 21h. 22m. 46s,

we see why the value he found for the inclination of the orbit was approximately its minimum value, and also why he was unable to detect the want of uniformity in the motion of the node.

127. To represent these changes in the position of the moon's orbit, Tycho made the following hypothesis.

Let ENF (fig. 18) be the ecliptic, K its pole, BAC a small circle, having also K for pole and at a distance from it equal to 5° 8'. Then, if we suppose the pole of the moon's orbit to move uniformly in the small circle and in the direction BAC, the node N, which is at 90° from both A and K, will retrograde uniformly on the ecliptic, and the inclination of the two orbits will be constant and equal to AK.

But instead of supposing the pole of the moon's orbit to be at A, let a small circle abcd be described with A as pole and a radius of 9' 30"; and suppose the pole of the moon's orbit to describe this small circle with double the velocity of the node in its synodical revolution which is accomplished in about 346 days, in such a manner that when the node is in syzygy the pole may be at a, the nearest point to K, and at c the most distant point when the node comes to quadrature, at b in the first and third octants, and at d in the second and fourth, so as to describe the small circle in about 173 days, the centre A of the small circle retrograding meanwhile with its uniform motion.

.......

=

By this method of representing the motion, we see that when node is in syzygy) the inclination (Ka = 5° 8′ — 94′ = 4° 584′, quadrature of the orbit is Kc=5° 8' 9' 5° 17', while at the octants it has its mean value Kb = Kd = KA = 5° 8'. Again, with respect to the motion of the node, since N is the pole of KaAc, it follows that when in syzygy and quadrature, the node occupied its mean place, in the first and third octants, the pole being at b, the node was behind its mean place by the angle bKA = (9' 30") cosec 5° 8' 1° 46', nearly,

=